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# On Some Ev-Degree and Ve-Degree Dependent Indices of Benes Network and Its Derived Classes

1 School of Software, Pingdingshan University, Pingdingshan, China

2 College of Computing and Information Technologies, National University, Manila, Philippines

3 Henan International Joint Laboratory for Multidimensional Topology and Carcinogenic Characteristics Analysis of Atmospheric Particulate Matter PM2.5, Pingdingshan, China

4 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan

* Corresponding Authors: Asfand Fahad. Email: ,

(This article belongs to this Special Issue: New Aspects of Computational Algorithms of Graphical Network in Fixed Point Theory)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1685-1699. https://doi.org/10.32604/cmes.2022.023563

**Received** 02 May 2022; **Accepted** 07 July 2022; **Issue published** 27 October 2022

## Abstract

One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks, Benes networks, Interconnection networks, and David-derived networks using graph theoretic parameters. The topological indices (*TIs*) have been widely used as graph invariants among various graph theoretic tools. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) need the use of

*TIs*. Different structure-based parameters, such as the degree and distance of vertices in graphs, contribute to the determination of the values of TIs. Among other recently introduced novelties, the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families. The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for dimensional Benes network and certain networks derived from it. In the end, a comparison between the values of the TIs for these networks has been presented through graphical tools.

## Keywords

A classic and helpful strategy is to attach various graphs to objects that may be an algebraic structure, a chemical structure of a drug, or a network, which assists in understanding certain features of the objects. Various graph parameters can be associated with the properties of the structure under examination which leads to a deeper study of its theory. These properties may include the algebraic properties of the zero divisor graphs and the physio chemical properties of the chemical structures. By adopting this strategy, several researchers studied different objects such as algebraic objects [1], pysio-chemical properties of chemical structures [2–4], drugs used for breast cancer treatment [5], and interconnection networks [6]. The analysis of networks, such as Butterfly network [7], Benes network [8,9], Interconnection network [6,10] and David-derived network [11] through similar approach, is one of the most recent developments in the field of graph theory.

The class of topological indices (TIs) is a significant class of parameters associated with graphs. Many TIs have been introduced and studied during the last fifty years. Among the class of degree dependent TIs, the Zagreb indices (ZIs) introduced in [12], vastly studied due to the ability to estimate the

Butterfly graphs are the associated graphs of Fast Fourier Transforms (FFT) networks which are especially effective in performing the FFT. The butterfly network is constructed by a sequence of switch stages and connector patterns that allow ′n′ different configurations. The ′n′ outputs should be connected to ′n′ inputs. Furthermore, the Benes network obtained by attaching back-to-back butterfly networks is known for permutation routing [8]. These networks are key multistage interconnection networks with appealing communication network topologies [9]. The graph associated to s-dimensional butterfly network consists of vertex set V with elements [v, i] in which v is an s-bit binary number representing the row of the node and

Let G(V,E) denotes a connected graph having V as a vertex set and E as an edge set. For

3 Ev-Degree Dependent Topological Indices for Bensen Networks and Its Derived Classes

In this section, we prove analytical formulae for the ev-degree dependent TIs for B(s), VCB(s) and HCB(s). The formulae have been established through partition of the vertex sets of B(s), VCB(s) and HCB(s) on the basis of ev-degree as shown in Tables 2–4. We start with the following theorem for B(s).

Theorem 3.1. For an s-dimensional Benes network B(s), we have:

(i)

(ii)

Proof.

By using Table 2, we compute the ev-degree based indices for Benes network as follows:

(i)

(ii)

Now, we continue to prove the ev-degree dependent TIs for VCB(s) in the next theorem.

Theorem 3.2. For VCB(s), the

(i)

(ii)

Proof. (i) From Table 3 and the definition of

(ii) From Table 3 and the definition of

We conclude the results of this section by proving the ev-degree dependent TIs for HCB(s):

Theorem 3.3. For HCB(s), the

(i)

(ii)

Proof. (i) From Table 4 and the definition of

which upon simplification gives the required result.

(ii) From Table 4 and the definition of

which upon simplification gives the required result.

Now, we present an example of the results proved in this section:

Example 3.1. By taking

4 Ve-Degree Dependent Topological Indices for Bensen Networks and Its Derived Classes

In this section, we develop formulae for the ve-degree dependent TIs for B(s), VCB(s) and HCB(s). The key to obtaining these formulae is to obtain partition the edge set of B(s), VCB(s) and HCB(s) on the basis of ve-degrees of the end vertices of each edge as shown in Tables 6–8.

Theorem 4.1. For B(s), the ve-degree dependent TIs

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Proof. (i) From Table 6 and the definition of

which upon simplification gives the required formula.

(ii) The Table 6 and the formula for

(iii) The Table 6 and the formula for

(iv) The Table 6 and the formula for

(v) The Table 6 and the formula for

(vi) The Table 6 and the formula for

(vii) The Table 6 and the formula for

(viii) Lastly, the Table 6 and the formula for

which upon simplification gives the required formula.

By using the Table 7 and the formulae of TIs defined in Table 1, we get the following results for the VCB(s).

Theorem 4.2. For VCB(s), ve-degree dependent TIs

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

By using the Table 8 and the formulae of TIs defined in Table 1, we get the following results for the HCB(s).

Theorem 4.3. For HCB(s), the ve-degree dependent TIs

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Now, we conclude the section by including the following example:

Example 4.1. By taking

We proceed further with our obtained formulae in previous section to study graphical patterns in the values of TIs of B(s), HCB(s) and HCB(s). In Figs. 4–6, the patterns of ZI(VE), RI(VE), ABC-I(VE), GA-(VE) and HI(VE) (on y-axis), where the value of s has been taken on x-axis, for B(s), HCB(s) and VCB(s) have been presented. All the figures show the rapid rise in the values of each TI for B(s), HCB(s) and HCB(s) with the rise in the value of s. The trends in Fig. 4 (L) show that the HCB(s) attains higher values of ZI(VE), whereas values of ZI(VE) for VCB(s) remain between B(s) and HCB(s). Similar trend for RI(VE) has been shown in Fig. 4 (R). In Fig. 5 (L), it can be seen that the values of ABC-I(VE) show different behaviours as in case of ZI(VE) and RI(VE). The ABC-I(VE) attains lowest values for HCB(s), whereas values for B(s) remain between VCB(s) and HCB(s). Furthermore, the trend for HI(EV) is shown in Fig. 5 (R). In the case of GA(VE), the values for B(s) remain the highest and the values for VCB(s) remain between the values of B(s) and HCB(s), see Fig. 6.

The study of newly formed networks is always a fascinating topic. Using B(s), several novel networks such as HCB(s) and VCB(s) have been defined through identifications in [33] and further investigated in [34]. Furthermore, the ev-degree and the ve-degree of these structures were not investigated yet. In the current work, we constructed the ev-degree based partition of the vertex set and the ve-degree based partition of the edge set for these networks. Through these partitions, we developed formulae for several ev-degree and ve-degree based TIs for B(s), HCB(s) and VCB(s) in terms of the parameter s. Further, we presented the comparative analysis of the values of ZI(VE), RI(VE), ABC-I(VE), GA-I(VE) and HI(VE) for B(s), HCB(s) and VCB(s). It is observed that similar patterns have been developed for ZI(VE) and RI(VE), whereas the other three TIs produce different trends.

Acknowledgement: The authors are thankful to their respective institutes.

Funding Statement: This work is partially supported by the National Natural Science Foundation of China (Grant No. 61702291); China Henan International Joint Laboratory for Multidimensional Topology and Carcinogenic Characteristics Analysis of Atmospheric Particulate Matter PM2.5.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

Wang, W., Arshad, H., Fahad, A., Javaid, I. (2023). On Some Ev-Degree and Ve-Degree Dependent Indices of Benes Network and Its Derived Classes.*CMES-Computer Modeling in Engineering & Sciences, 135(2)*, 1685–1699.